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Inaugural Dissertation INAUGURAL DISSERTATION zur Erlangung der Doktorw¨urde der Naturwissenschaftlich-Mathematischen Gesamtfakult¨at der Ruprecht-Karls-Universit¨at Heidelberg vorgelegt von M.Sc. Yamidt Berm´udez Tob´on aus Montebello (Ant.), Kolumbien Tag der m¨undlichen Pr¨ufung: Thema An efficient algorithm to compute an elliptic curve from a corresponding function field automorphic form Gutachter: Prof. Dr. Gebhard B¨ockle Acknowledgements I thank my supervisor Prof. Dr. Gebhard B¨ockle for his guidance and support. Also I would like to thank to Dr. Juan Cervi˜no, he provided sound support and was always willing to help whenever I asked, during the whole core of this work. We had a lot of interesting and enthusiastic discussions which often contained useful ideas. Without their help this thesis would not have been possible. During this work I was supported by the DAAD for three and one half years and the DFG project Nr D.110200/10.005 from 01.01.2013 to 30.04.2014. Abstract Elliptic modular forms of weight 2 and elliptic modular curves are strongly related. In the rank-2 Drinfeld module situation, we have still modular curves that can be described analytically through Drinfeld modular forms. In [GR96] Gekeler and Reversat prove how the results of [Dri74] can be used to construct the analytic uniformization of the elliptic curve attached to a given automorphic form. In [Lon02] Longhi, building on ideas of Darmon, defines a multiplicative integral that theoretically allows to find the corresponding Tate parameter. In this thesis we develop and present a polynomial time algorithm to compute the integral proposed by Longhi. Also we devised a method to find a rational equation of the corresponding representative for the isogeny class. Zusammenfassung Elliptische Modulformen von Gewicht 2 und elliptische Modulkurven stehen in enger Verbindung. Im Fall eines Drinfeld-Moduls von Rang 2 haben wir noch Modulkurven, die durch Drifeldsche Modulformen analytisch beschrieben wer- den k¨onnen. Gekeler und Reversat [GR96] beweisen, wie die Ergebnisse von [Dri74] genutzt werden k¨onnen, um die analytische Uniformisierung der, einer gegeben automorphen Form eingeordneten elliptischen Kurve, zu konstruieren. Auf Darmons Ideen aufbauen definiert Lonhi [Lon02] ein multiplikatives Inte- gral, das es erlaubt, den entsprechenden Tate-Parameter zu finden. In der vor- liegenden Arbeit wird ein Polynomialzeitalgorithmus entwickelt und vorgestellt, um das von Longhi vogeschlagene Integral zu berechnen. Ausserdem wird eine Methode entwickelt, mit der eine rationale Gleichumg der entsprechen- den Vertreter der Isogenieklasse gefunden werden kann. Contents Nomenclature vi 1 Introduction 1 2 Background 7 2.1 Notation..................................... 7 2.2 Notionsfromgraphtheory........................... 7 2.3 TheDrinfeldupperhalfplane ......................... 10 2.4 TheBruhat-Titstree .............................. 11 2.5 Endsofthetree................................. 13 2.6 Thereductionmap ............................... 17 2.7 Drinfeldmodularcurves ............................ 19 2.8 Thequotientgraph............................... 20 2.9 Harmoniccocycles ............................... 21 2.10 Thetafunctionsforarithmeticgroups . ... 24 2.10.1 Theta functions and harmonic cocycles . 25 iii Contents 2.11Heckeoperators................................. 26 2.12 Application to the Shimura-Taniyama-Weil uniformization . ... 27 3 Integration, Theta function and uniformizations 31 3.1 Integration.................................... 31 3.1.1 Measuresandharmoniccocycles. 32 3.1.2 The integral over ∂Ω.......................... 33 3.1.3 Changeofvariables........................... 33 3.2 Thetafunction ................................. 34 3.3 Complexuniformization ............................ 35 3.4 p-adicUniformization.............................. 38 4 The Algorithm 45 4.1 Motivation.................................... 45 4.2 Elementaryfunctions.............................. 46 4.3 AHeckeoperator ................................ 50 4.4 The change of variables and calculation of the integral . ... 61 4.4.1 Choosing the z0 ............................. 62 4.4.2 Thepartitionoftheborder . .. .. 62 4.4.3 Thechangeofvariables......................... 63 5 Applications and examples 69 5.1 Ellipticcurves.................................. 69 5.2 Supersingular elliptic curves . 78 5.3 Elliptic curves over C andEisensteinseries . 79 iv Contents 5.4 Reduction modulo p ofmodularforms .................... 84 5.5 TheTateCurve................................. 87 5.6 ObtainingtheTateparameter . .. .. 90 5.7 Obtainingequationsforthecurves . 91 5.7.1 Elliptic curves in characteristic 2 and 3 . 92 5.7.2 Elliptic curves over characteristic p> 3................ 102 A Algorithms for the Quotient graph 113 A.1 Computational complexity of mathematical operations . .... 113 A.2 Representatives for the edges of Γ T .................... 115 0 \ A.3 Lifting cycles to T ................................ 124 A.4 Findingtherepresentative . .. .. 126 B Algorithms for the table 129 B.1 Algorithms for the calculation of the table . 129 B.2 Algorithms for the calculation of the integral . 141 C Tables 149 C.1 Preliminaries .................................. 149 C.2 Table for degree 3 over F3 ........................... 150 C.3 Table for degree 4 over F3 ........................... 150 C.4 Table for degree 3 over F5 ........................... 152 C.5 Table for non-primes of degree 4 over F5 ................... 153 C.6 Table for primes of degree 4 over F5 ...................... 166 C.7 Table for primes of degree 5 over F5 ...................... 168 v Contents References 174 vi 1. Introduction The theory of modular forms and their relation to the arithmetic of elliptic curves is a central subject in modern mathematics, where most diverse branches of mathematics come together: complex analysis, algebraic geometry, representation theory, algebra and number theory. One of the most exciting and widely mathematical discoveries is the proof, by Andrew Wiles, of “Fermat last’s theorem”, its solution draws an incredible range of modern mathematics, which is precisely the relation between modular forms and elliptic curves. There are a number of analogies between on the one hand, the integers Z and the rational numbers Q and on the other hand, Fq[T ] and its field of fractions Fq(T ). Frequently a problem posed in number fields or, in other words, in finite extensions of Q, admits an analogous problem in function fields, and the other way around. For example, since the appearance of Drinfeld’s work [Dri74], we know that all elliptic curves which are semistable at the place are modular, that is, they appear as a factor of the Jacobian of a Drinfeld ∞ modular curve. We are interested in number theory over function fields, particularly in elliptic curves over Fq(T ). In order to get a better idea of the function field case, it is worth to start with a short description of the classical case, that is, over the rational numbers Q. Let f : H C be −→ a cuspidal modular form of weight two for the Hecke congruence subgroup Γ0(N) which is also a new eigenform with rational Hecke eigenvalues, we call it for short “a Q-rational newform” of level N. From the Eichler-Shimura theory, with f one can associate an elliptic curve E with conductor N and a morphism defined over Q form the modular curve X0(N) to the elliptic curve E. Such an elliptic curve E is called a Weil curve. On the other hand, since the 60’s (after the work of Shimura, Taniyama, and Weil) emerged 1 1. Introduction the conjecture that all elliptic curves over Q (up to isogeny) should be obtainable from the Eichler-Shimura construction. The conjecture known as the Shimura-Taniyama-Weil conjecture, is now a theorem called the modularity theorem [BCDT]. Basically it states that there are canonical bijections between the sets of 1) Normalized Q-rational newforms f of level N with rational Hecke eigenvalues; 2) One dimensional isogeny factors of the new part of the Jacobian of the modular curve X0(N); 3) Isogeny classes of elliptic curves E over Q with conductor N. The previous correspondence yields an effective method to determine all elliptic curves E/Q with a given conductor N, since the modular parametrization is explicitly and effectively computable (c.f. [Cre97]). For tables and numerical results see ([Ibid., Ch. 4]). For some applications it is convenient to consider other kind of parametrizations instead of the modular one, for example the Shimura parametrization, introdued in [BC91] and [BD98]. Let E be an elliptic curve of conductor N and suppose that N is square free and + factorizes as N = N −N where N − has a even number of factors. Then there exits a parametrization of E by the Shimura curve X − + (cf. 3.4), that is a non constant mor- N N § phism from the Jacobian of the Shimura curve to E. However the lack of q-expansions for modular forms on non-split quaternion algebras, forces one to consider p-adic uniformiza- tions of E by certain discrete arithmetic subgroups of SL2(Qp) at the primes p dividing N −. The modular forms considered here may be regarded as functions on oriented edges of certain Bruhat-Tits tree T (cf. 3.4), called harmonic cocycles, which can be identified § 1 with measures on P (Qp). Using these measures Bertolini and Darmon are able to define 1 a multiplicative p-adic integral defined over P (Qp), which theoretically gives rise to the modular parametrization of the Tate curve attached to a given harmonic cocycle. In [Gre06], Greenberg gives an algorithm, running
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